ntroduce some simple general theory to allow us to talk about notions of higher-type computable functional. The following definitions (with minor variations) appear frequently in the literature. Definition 1.1 (Weak partial type structures) A weak partial type structure, or weak PTS A [over a set X], consists of the following data: . for each type #, a set A # of elements of type # [equipped with a canonical bijection A 0 # = X], . for each #, # , a partial application function ## : A ### A # # A # . We usually omit type subscripts from application operations, and often write x y simply as xy. By convention, w

Abstract. In [6] J. Laird has shown that an infinitary sequential extension of PCF has a fully abstract model in his category of locally boolean domains (introduced in [8]). In this paper we introduce an extension SPCF ∞ of his language by recursive types and show that it is universal for its model in locally boolean domains. Finally we consider an infinitary target language CPS ∞ for (the) CPS translation (of [16]) and show that it is universal for a model in locally boolean domains which is constructed like Dana Scott’s D ∞ where D = 1

Abstract. We use a fully abstract denotational model to show that nested function calls and recursive definitions can be eliminated from SPCF (a typed functional language with simple non-local control operators) without losing expressiveness. We describe — via simple typing rules — an affine fragment of SPCF in which function nesting and recursion (other than iteration) are not permitted. We prove that this affine fragment is fully expressive in the sense that every term of SPCF is observationally equivalent to an affine term. Our proof is based on the observation of Longley — already used to prove universality and full abstraction results for models of SPCF — that every type of SPCF is a retract of a first-order type. We describe retractions of this kind which are definable in the affine fragment. This allows us to transform an arbitrary SPCF term into an affine one by mapping it to a first-order term, obtaining an (affine) normal form, and then projecting back to the original type. In the case of finitary SPCF, the retraction is based on a simple induction, which yields bounds for the size of the resulting term. In the infinitary case, it is based on an analysis of the relationship between SPCF definable functions and strategies for computing them sequentially. 1

real-world applications (e.g. [EHM + 99, Buh95]). Moreover, aspects of ML such as strong typing and the exceptions system have significantly influenced the design of languages such as Java [GJS96], and it seems likely that future systems languages will incorporate many of these features [Mac00]. Regarding the second requirement, even before the definition of ML had fully taken shape, the LCF system [GMW78] provided a program logic for a rather restricted fragment of the language. Subsequent research has sought to build on the definition in order to support formal reasoning about programs. Most notably, the Extended ML project [KST97] resulted in a formal language for specifying program properties, but the complexity of this language prohibited the development of useful proof rules. A di#erent approach has been pursued by Elsa Gunter et al [GV94], who have formalized the definition of ML within the HOL theorem prover; this has proved useful for metatheo

Abstract. We formalize SℓPCF, namely a programming language which is able to represent linear function between coherence spaces. We give an interpretation of SℓPCF into the model of linear coherence spaces and we show that such semantics is fully abstract with respect to it. SℓPCF is not syntactically linear, namely its programs can contain the same variable more than once. Last, we address the universality problem. 1

We give an overview of some recently discovered realizability models that embody notions of sequential computation, due mainly to Abramsky, Nickau, Ong, Streicher, van Oosten and the author. Some of these models give rise to fully abstract models of PCF; others give rise to the type structure of sequentially realizable functionals, also known as the strongly stable functionals of Bucciarelli and Ehrhard. Our purpose is to give an accessible introduction to this area of research, and to collect together in one place the definitions of these new models. We give some precise definitions, examples and statements of results, but no full proofs. Preface Over the last two years, researchers in various places (principally Abramsky, Nickau, Ong, Streicher, van Oosten and the present author) have come up with a number of new realizability models that embody some notion of "sequential" computation. Many of these give rise to fully abstract and universal models for PCF and related languages. Alth...

ped strategy (in the sense of the AJM model of Abramsky et. al.) for d. Accordingly, the realisability model over A eff wb , the effective well--bracketed strategies, is even universal in the sense that all elements of the model appear as denotations of PCF terms. In a future version of [9] there will also be included the discussion of other pca's of game--theoretic nature giving rise to fully abstract and universal models which originate from S. Abramsky's work on game semantics for classical linear logic. Independently, fully abstract realisability models for PCF have been constructed by Marz, Rohr and Streicher in [14] using as underlying pca's term models for untyped --calculus with arithmetic. In this case the proof is not via the AJM game model but instead makes use of the category SD of sequential domains as described in [13] originating from a reformulation and generalisation of [18]. As describ

5.1 Definition and Examples 5.1.1 Definition and Definability Results A tripos is a weak tripos with disjunction which has a (weak) generic object. Explicitly we define:

Abstract not found

Interpretation is an implicit part of today’s programming; it has great power but is overused and has significant costs. For example, interpreters are typically significantly hard to understand and hard to reason about. The methodology of “Totally Functional Programming” (TFP) is a reasoned attempt to redress the problem of interpretation. It incorporates an awareness of the undesirability of interpretation with observations that definitions and a certain style of programming appear to offer alternatives to it. Application of TFP is expected to lead to a number of significant outcomes, theoretical as well as practical. Primary among these are novel programming languages to lessen or eliminate the use of interpretation in programming, leading to better-quality software. However, TFP contains a number of lacunae in its current formulation, which hinder development of these outcomes. Among others, formal semantics and type-systems for TFP languages are yet to be discovered, the means to reduce interpretation in programs is to be determined, and a detailed explication is needed of interpretation, definition, and the differences between the two. Most important of all however is the need to develop a complete understanding of the nature of interpretation. In this work, suitable type-systems for TFP languages are identified, and guidance given regarding the construction of appropriate formal semantics. Techniques, based around the ‘fold’ operator, are identified and developed for modifying programs so as to reduce the amount of interpretation they contain. Interpretation as a means of language-extension is also investigated. v Finally, the nature of interpretation is considered. Numerous hypotheses relating to it considered in detail. Combining the results of those analyses with discoveries from elsewhere in this work leads to the proposal that interpretation is not, in fact, symbol-based computation, but is in fact something more fundamental: computation that varies with input. We discuss in detail various implications of this characterisation, including its practical application. An often more-useful property, ‘inherent interpretiveness’, is also motivated and discussed in depth. Overall, our inquiries act to give conceptual and theoretical foundations for practical TFP.